Question 8 of 28beginner🔧 ApplyNumerical1 mark

Express the following complex number in the form a+iba + ib: (1−i)4(1 - i)^4

Correct Answer

-4

Exercise: EXERCISE 4.1 | Q: 8 | (Page 8)
For More Understanding

Explanation

To express the complex number (1−i)4(1 - i)^4 in the form a+iba + ib, we expand the expression using the binomial theorem, similar to the method shown in Example 3 of the provided text. We use the properties of ii where i2=−1i^2 = -1, which leads to i3=−ii^3 = -i and i4=1i^4 = 1 (as stated in the Summary for k=1k=1).

Solution Steps

  1. Step 1: Expand (1−i)4(1 - i)^4 using the binomial expansion (a−b)4=a4−4a3b+6a2b2−4ab3+b4(a-b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4.

  2. (1−i)4=14−4(1)3(i)+6(1)2(i)2−4(1)(i)3+(i)4(1 - i)^4 = 1^4 - 4(1)^3(i) + 6(1)^2(i)^2 - 4(1)(i)^3 + (i)^4

  3. Step 2: Simplify the powers of ii using i2=−1i^2 = -1, i3=−ii^3 = -i, and i4=1i^4 = 1.

  4. =1−4i+6(−1)−4(−i)+1= 1 - 4i + 6(-1) - 4(-i) + 1

  5. Step 3: Group the real and imaginary parts.

  6. =(1−6+1)+i(−4+4)= (1 - 6 + 1) + i(-4 + 4)

  7. Step 4: Perform the final addition to get the form a+iba + ib.

  8. =−4+0i= -4 + 0i

  9. Answer: −4-4

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